Classical Density Functional Theorypruess/publications/dft.pdf · External potential is a unique functional of the density proﬁle Solve self-consistency equation (i.e. ﬁnd root,

IntroductioncDFT in general

Application to interfaces

Classical Density Functional TheoryTowards interfaces in crystalline materials

Gunnar Pruessner and Adrian Sutton

Department of PhysicsImperial College London

INCEMS M12 meeting, Imperial College London, August 2006

G. Pruessner and A. Sutton (Imperial) Classical Density Functional Theory INCEMS M12, 08/2006 1 / 16



Outline

1 IntroductionAimsRoots

2 cDFT in generalGeneral considerationsIdeal SystemPerturbation Theory

3 Application to interfacescDFT for interfaces: Haymet and OxtobyApplication to IGFs




AimsRoots

Outline







AimsRoots

IntroductioncDFT: What for?

Aims of classical density functional theoryDetermine thermodynamic properties: (surface) free energy,density profiles, phases, potentials, etc. without simulation orexperimentsDetermine (few) relevant degrees of freedomMinimal input (here: direct n-point correlation function)Controlled approximation scheme (perturbation theory. . . )Here: Given two crystalline orientations at x = ±∞, what is thedensity profile in between? (notice: structured walls!)




AimsRoots

IntroductioncDFT: Based on what?

Theoretical basis of classical density functional theoryGiven a stable (metastable?) thermodynamic system . . .. . . write down a (mock-) free energy (grand potential)External potential is a unique functional of the density profileSolve self-consistency equation (i.e. find root, i.e. minimise mockpotential)Relation to electronic DFT: Ground state wave-function is aunique functional of density profile




General considerationsIdeal SystemPerturbation Theory

Outline








The key equation of DFTDFTGiven the full free energy functional F[ρ] of the system, the uniquepotential u(~r) (intrinsic chemical potential) at given density profile ρ(~r),or one possible solution ρ(~r) at given u(~r) is found by minimising

W[ρ, u] = F[ρ] −

∫ddr u(~r)ρ(~r)

with respect to ρ (i.e. δδρW ≡ 0) which is equivalent to solving

u(~r)[ρ] = u(~r)

plus technicalities (LHS: equilibrium potential, RHS: imposedpotential). If ρ is the equilibrium profile for u then W[ρ, u] is the grandpotential.





Flavours of DFT

Impose that the free energy has square gradient formDensity expansion (expansion about bulk; HNC [here] and PYclosure)LDA: F =

∫ddr f (ρ)

Expansion about bulk; van der WaalsWDA: F = . . . +

∫ddrρ(~r)f (~r)[ρ] (Tarazona Mark I, Mark II,

Curtin-Ashcroft, . . . )Rosenfeld’s fundamental-measure theory. . .





General considerations I

Hamiltonian of an N-particle system:

HN = VN(~r1, . . . ,~rN) +

N∑i

(p2

i

2m+ U(ri)

)Grand canonical partition sum:

Z =∑

N

1N!

eβµNΛ−Nd∫

ddp1ddx1 . . .

∫ddpNddxN e−βHN

= Tr1

N!eβµNΛ−Nde−βHN

where Λ is the phase space volume element. Integrate out(Gaussian) pi’s (. . .

√2mπ/β) and absorb into Λd → Λ.





General considerations II

Introduce density operator:

ρN(~r;~r1, . . .~rN) =

N∑i

δ(~r −~ri)

and write∑

i U(ri) − µN =∫

ddrρN(U(~r) − µ).Dimensionless (intrinsic chemical) potential u(~r) = βµ − βU(~r) sothat

eβµNe−β∑

i U(~ri) = e∫

ddrρN(~r;...)u(~r)

Grand canonical partition sum now

Z = Tr1

N!Λ−Ne−βVN(...)e

∫ddrρNu





General considerations IIIGrand potential W = − ln Z

so thatρ(~r) = 〈ρ(~r; . . .)〉 = −

δ

δu(~r)W

and 〈N〉 = −β−1 ddµW

Grand potential is a functional of the external potential,W = W(β, [u(~r)]). Its Legendre transform

F = F(β, [ρ(~r)]) = W +

∫ddr u(~r)ρ(~r)

is the free energy.Straight forward to show that

δ

δρ(~r)F = u(~r)





Ideal System I

Ideal system: VN ≡ 0, so partition sum becomes

Zid = Tr1

N!Λ−N exp

(∫ddr ρN(~r; . . .) u(~r)

)Because of the missing interaction, the integral factorises andbecomes just (∫

ddr eu(~r))N

so that

Zid =∑

N

1N!

Λ−N(∫

ddreu(~r))N

= exp(

Λ−1∫

ddr eu(~r))





Ideal System II

From Z = exp(Λ−1∫

ddr exp(u)) we have the grand potential:

Wid[u(~r)] = −Λ−1∫

ddr exp(u(~r))

and from ρ = − δδu W we have the barometric formula:

ρ(~r) = Λ−1 exp(u(~r))

Using this relation between ρ and u one can perform the Legendretransform (F = W +

∫dr ρu) to find

Fid[ρ(~r)] =

∫ddr (ln(ρΛ) − 1) ρ





Ideal System IIINow define the excess free energy Φ for an interacting system as

Φ[ρ(~r)] = Fid[ρ(~r)] − F[ρ(~r)]

where Fid[ρ(~r)] is the free energy for a given density profile,assuming that the system is ideal.Use δ

δρFid = ln(ρΛ) and δδρF = u to show that

C(~r) ≡ δ

δρ(~r)Φ = ln(ρ(~r)Λ) − u(~r)

which is the effective one particle potential: It is the additionalexternal potential needed in an ideal system, for it to “mimic” aparticular density profile ρ found in an interacting system

ρ(~r) = Λ−1eu(~r)+C(~r)





Perturbation Theory: cDFT

Standard technique in cDFT: Write down a perturbative expansion ofthe effective one particle potential about some reference system:

C(~r) = C(1)0 (~r) +

∫ddr1C(2)

0 (~r,~r1) (ρ(~r1) − ρ0(~r1)) + . . .

which, together with

C(~r) = ln(ρ(~r)Λ) − u(~r)

immediately gives rise to a self-consistency equation.Connection to the real world via the n-point direct correlation functionsC(n)

0 (~r1, . . . ,~rn) of the reference system.




cDFT for interfaces: Haymet and OxtobyApplication to IGFs

Outline








cDFT for interfacesHaymet and Oxtoby I — The uniform system

Consider a continuous family (index~s) of uniform (uniform Fouriercoefficient, i.e. periodic) systems

ρ(~r;~s) = ρ0

(1 +

∑n

µn(~s)eı~kn~r

)Every member has its own external potential

U(~r;~s) =∑

n

Un(~s)eı~kn~r

Expand uniform systems about the bulk liquid with bulk density ρ0one particle potential Cl and chemical potential µ, so

Cu(~r;~s) = Cl +

∫ddr ′ C(2)(~r −~r ′)(ρ(~r ′;~s) − ρ0)

with u(~r;~s) = ln(ρ(~r;~s)Λ) − Cu(~r;~s).G. Pruessner and A. Sutton (Imperial) Classical Density Functional Theory INCEMS M12, 08/2006 13 / 16




cDFT for interfacesHaymet and Oxtoby I — Potential

Now use the infinite bulk liquid

βµ = ln(ρ0Λ) − Cl

to derive

The potential

−βU(~r;~s) = ln(ρ(~r;~s)/ρ0) −

∫ddr ′ C(2)(~r −~r ′)

(ρ(~r ′;~s) − ρ0

)





cDFT for interfacesHaymet and Oxtoby II — The interface

Impose that the family collectively represents a system withvarying density profile – the interface:

ρi(~r) = ρ(~r;~r) = ρ0

(1 +

∑n

µn(~r)eı~kn~r

)







ρi(~r) = ρ(~r;~r) = ρ0

(1 +

∑n

µn(~r)eı~kn~r

)







ρi(~r) = ρ(~r;~r) = ρ0

(1 +

∑n

µn(~r)eı~kn~r

)







ρi(~r) = ρ(~r;~r) = ρ0

(1 +

∑n

µn(~r)eı~kn~r

)







ρi(~r) = ρ(~r;~r) = ρ0

(1 +

∑n

µn(~r)eı~kn~r

)







ρi(~r) = ρ(~r;~r) = ρ0

(1 +

∑n

µn(~r)eı~kn~r

)







ρi(~r) = ρ(~r;~r) = ρ0

(1 +

∑n

µn(~r)eı~kn~r

)







ρi(~r) = ρ(~r;~r) = ρ0

(1 +

∑n

µn(~r)eı~kn~r

)







ρi(~r) = ρ(~r;~r) = ρ0

(1 +

∑n

µn(~r)eı~kn~r

)What is actually imposed? No potential for interface:

βµ = ln(ρi(~r)Λ) − Ci(~r)

with the effective one particle potential to be expanded about theinfinite liquid: Ci(~r) = Cl +

∫ddr ′ C(2)(~r −~r ′)(ρi(~r ′) − ρ0)

Resulting relation between Ci and Cu:

Ci(~r) − Cu(~r;~s) =

∫ddr ′C(2)(~r −~r ′)(ρ(~r ′;~r ′) − ρ(~r ′;~s))





cDFT for interfacesHaymet and Oxtoby III — Equation of motion

Relation between Ci and Cu must be consistent with

βµ = ln(ρ(~r;~r)Λ) − Ci(~r) (1)βµ − βU(~r;~s) = ln(ρ(~r;~s)Λ) − Cu(~r) (2)

Evaluate at~s = ~r

−βU(~r;~r) =

∫ddr ′ C(2)(~r −~r ′)(ρ(~r ′;~r ′) − ρ(~r ′;~r))

This is the external potential to be applied in the uniform system,so that the resulting family of density profiles gives rise to aninterface profile with vanishing potential.Note: if ρ(~r;~s) ≡ ρ∗(~r) independent of~s, then all uniform systemsare the same, no potential, identical to the interface.





cDFT for interfacesHaymet and Oxtoby III — Equation of motion

Self-consistency above fixes U only at~s = ~r.Make a convenient choice:

The equation of motion

−βU(~r;~s) =

∫ddr ′ C(2)(~r −~r ′)

(ρ(~r ′;~r ′ +~s −~r) − ρ(~r ′;~s)

)Advantage: Taylor-expand bracket and plug in Fourier-sum forρ(~r;~s). . .





cDFT for interfacesHaymet and Oxtoby IV — The central result

Equation of motion for an interface across z:

−βUn(z) = −ρ0V{

µ ′n(z)ıC

(2)′(~kn) +12µ ′′

n (z)C(2)′′(~kn) + . . .

}Potential:

−βUn(z) = V−1∫

Vddr e−ı~kn~r ln

(1 +

∑n

µn(z)eı~kn~r

)−ρ0VC(2)(~kn)µn(z)





Application to IGFsApply appropriate boundary conditions for µn(z → ±∞)

Find appropriate parametrisation capable of capturing both sidesof the interfaceFind direct correlation function for bulk-liquid phaseSolve self-consistency (i.e. integrate equation of motion) . . . Done!























Problems with O&H’s original approachParametrisation

Fourier sum makes physical sense and makes the convolutionfactoriseParametrisation must be capable to capture different translationsand orientations on both sides (Fourier transform breakssymmetry)Parametrisation must make sure that a stable, uniformconfiguration is equally stable in any orientation and translationAll possible degrees of freedom should (but cannot) be accessible– even those that break symmetry (numerical inaccuracies help)Solid phase not necessarily stable (reference system: liquid)





























What to do next?

Find a parametrisation that keeps the solid stable in anytranslation and orientationExpand about a (stable) solid, i.e. take the direct correlationfunction from the stable solidC(2)′ and C(2)′′ à la Debye WallerConcentrate on gaps and Σ-boundaries





What to do next?






What to do next?






What to do next?



Documents

Classical Density Functional Theorypruess/publications/dft.pdf · External potential is a unique functional of the density proﬁle Solve self-consistency equation (i.e. ﬁnd root,